Sampling in spaces of entire functions of exponential type in Cn+1

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–Pólya type inequality.